A minimal Lagrangian \(\widetilde{\Sigma }\)  in \(\mathbb {H}^{2} \times \mathbb {H}^{2}\)  is a minimal surface which is Lagrangian with respect to the symplectic form \(\omega \oplus -\omega \) , where \(\omega \)  is the standard Kähler form on \(\mathbb {H}^{2}\) . Any \(\rho \in \mathrm {Max}(S)\)  acts on \(\mathbb {H}^{2} \times \mathbb {H}^{2}\) , and Schoen [1]} has shown to each such \(\rho \) , there is a unique \(\rho \) -equivariant minimal Lagrangian \(\widetilde{\Sigma }_{\rho }\)  in \(\mathbb {H}^{2} \times \mathbb {H}^{2}\) , thereby providing a geometric interpretation to representations in \(\mathrm {Max}(S)\) . The second author [2]} has studied the degeneration of these minimal Lagrangians, and has shown that one may interpret the space \(\mathbb {P}\mathrm {Core}(\mathcal {T},\mathcal {T})\)  as the boundary of the maximal component \(\mathrm {Max}(S)\) .
